Esquisse d'un Programme
Encyclopedia
"Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck
. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal.
, or "children's drawings", and "anabelian geometry
", that are contained in this manuscript continue to inspire research.
Suggested further reading for the interested mathematical reader is provided
in the References section.
in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups
, which is now known as the theory of Galois group
s; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups. To begin with, Alexander Grothendieck stated in his proposal: "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group
which respects certain structures that are essential to this group." This fundamental, Galois group theory in mathematics has been considerably expanded, at first to groupoids- as proposed in Alexander Grothendieck's Esquisse d' un Programme (EdP)- and now already partially carried out for groupoids; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's IHÉS seminars (SGA1 to SGA4) held in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's Descent Theory. The notion of motive
has also been pursued actively. This was developed into the motivic Galois group, Grothendieck topology
and Grothendieck category
. Such developments were recently extended in algebraic topology
via representable functor
s and the fundamental groupoid functor.
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal.
Brief history
Submitted in 1984, the Esquisse d'un Programme was a successful proposal submitted by Alexander Grothendieck for a position at the Centre National de la Recherche Scientifique, which Grothendieck held from 1984 till 1988. This proposal was however not formally published until 1997, because the author "could not be found, much less his permission requested". The outlines of dessins d'enfantsDessin d'enfant
In mathematics, a dessin d'enfant is a type of graph drawing used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.Intuitively, a dessin d'enfant is simply a graph, with its vertices colored alternating black and...
, or "children's drawings", and "anabelian geometry
Anabelian geometry
Anabelian geometry is a proposed theory in mathematics, describing the way the algebraic fundamental group G of an algebraic variety V, or some related geometric object, determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian...
", that are contained in this manuscript continue to inspire research.
Abstract of Grothendieck's programme
("Sommaire")- 1. The Proposal and enterpise ("Envoi").
- 2. "TeichmüllerOswald TeichmüllerOswald Teichmüller was a German mathematician who introduced quasiconformal mappings and differential geometric methods into complex analysis.-Life:...
's Lego-game and the Galois groupGalois groupIn mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
of Q over Q" ("Un jeu de “Lego-Teichmüller” et le groupe de Galois de Q sur Q"). - 3. Number fields associated with "dessin d'enfants". ("Corps de nombres associés à un dessin d’enfant").
- 4. Regular polyhedraRegular polyhedronA regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
over finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s ("Polyèdres réguliers sur les corps finis"). - 5. General topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
or a 'Moderated Topology' ("Haro sur la topologie dite 'générale', et réflexions heuristiques vers une topologie dite 'modérée"). - 6. Differentiable theories and moderated theories ("Théories différentiables" (à la Nash) et “théories modérées").
- 7. Pursuing Stacks ("À la Poursuite des Champs"}.
- 8. Two-dimensional geometry ("Digressions de géométrie bidimensionnelle".
- 9. Summary of proposed studies ("Bilan d’une activité enseignante").
- 10. Epilogue.
- Notes
Suggested further reading for the interested mathematical reader is provided
in the References section.
Extensions of Galois's theory for groups: Galois groupoids, categories and functors
Galois has developed a powerful, fundamental algebraic theoryAlgebraic theory
In mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.Saying that...
in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, which is now known as the theory of Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups. To begin with, Alexander Grothendieck stated in his proposal: "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group
Pro-finite group
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
which respects certain structures that are essential to this group." This fundamental, Galois group theory in mathematics has been considerably expanded, at first to groupoids- as proposed in Alexander Grothendieck's Esquisse d' un Programme (EdP)- and now already partially carried out for groupoids; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's IHÉS seminars (SGA1 to SGA4) held in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's Descent Theory. The notion of motive
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
has also been pursued actively. This was developed into the motivic Galois group, Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
and Grothendieck category
. Such developments were recently extended in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
via representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s and the fundamental groupoid functor.
See also
- Grothendieck's Galois theoryGrothendieck's Galois theoryIn mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry...
- Grothendieck's Séminaire de géométrie algébriqueGrothendieck's Séminaire de géométrie algébriqueIn mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
Related works by Alexander Grothendieck
- Alexander GrothendieckAlexander GrothendieckAlexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
. 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente, Lecture Notes in Math. 224, Springer-Verlag: Berlin.
- Alexander Grothendieck. 1957, Sur quelque point d'algébre homologique. , Tohoku Math. J., 9: 119-221.
- Alexander Grothendieck and Jean DieudonnéJean DieudonnéJean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
.: 1960, Éléments de géométrie algébriqueÉléments de géométrie algébriqueThe Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...
., Publ. Inst. des Hautes Études Scientifiques, (IHÉS), 4.
- Alexander Grothendieck et al.,1971. Séminaire de Géométrie Algébrique du Bois-MarieGrothendieck's Séminaire de géométrie algébriqueIn mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
, Vol. 1-7, Berlin: Springer-Verlag.
- Alexander Grothendieck. 1962. Séminaires en Géométrie Algébrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohèrents et Théorèmes de Lefschetz Locaux et Globaux. , pp. 287. (with an additional contributed exposé by Mme. Michele Raynaud). (Typewritten manuscript available in French; see also a brief summary in English References Cited:
- Jean-Pierre SerreJean-Pierre SerreJean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
. 1964. Cohomologie Galoisienne, Springer-Verlag: Berlin. - J. L. Verdier. 1965. Algèbre homologiques et Catégories derivées. North Holland Publ. Cie).
- Jean-Pierre Serre
- Alexander Grothendieck. 1957, Sur quelques points d'algèbre homologique, Tohoku Mathematics Journal, 9, 119-221.
- Alexander Grothendieck et al. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results. AG4 is freely available in French; also available is an extensive Abstract in English.
- Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), finally published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University PressCambridge University PressCambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
, 1997, pp. 5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .
- Alexander Grothendieck, "La longue marche in à travers la théorie de Galois." = "The Long March Towards/Across the Theory of Galois", 1981 manuscript, University of MontpellierUniversity of MontpellierThe University of Montpellier was a French university in Montpellier in the Languedoc-Roussillon région of the south of France. Its present-day successor universities are the University of Montpellier 1, Montpellier 2 University and Paul Valéry University, Montpellier III.-History:The university...
preprint series 1996, edited by J. Malgoire.
External links
- Fundamental Groupoid Functors, Planet Physics.
- The best rejected proposal ever, Never Ending Books, Lieven le Bruyn